Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x-7y &= 9 \\ 6x+9y &= -9\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $6$ and the bottom equation by $5$ $\begin{align*}-30x-42y &= 54\\ 30x+45y &= -45\end{align*}$ Add the top and bottom equations. $3y = 9$ Divide both sides by $3$ and reduce as necessary. $y = 3$ Substitute $3$ for $y$ in the top equation. $-5x-7( 3) = 9$ $-5x-21 = 9$ $-5x = 30$ $x = -6$ The solution is $\enspace x = -6, \enspace y = 3$.